Average damage of critical hits

Question

I am looking for a better understanding of the following formula.

In games with Crit Rate and Crit Damage percentages (like Genshin impact or Honkai Star rail to quote a few) the expected average damage of criticals is calculated as follow according to the wiki I could read : Base Damage * (1+CritRate*CritDamage)

However it seems inconsistent with how I understand the logic which is that the probability of hitting a Crit (Crit Rate) should be multiplied by the Critical Damage + 1 because Critical damage in particular when it reaches for instance 100% is actually is actually equal to doubling the damage (200%) thus the +1. Meaning this formula instead : Base Damage * (Crit Rate * (1+ Crit Damage))

I'd like if possible to understand which ones is correct.

Answer 1

Let's define the variables:

• Base Damage : $b$
• Critical Damage : $c$
• Critical Rate : $r$

The chance of landing a critical hit is $r$. When the hit is critical, the damage is $b\times [1+c]$. When the hit is not critical, the damage is simply $b$.

The expected value (average damage over a large number of hits) is given by:

$$d = [1-r]\times b + r\times b\times[1+c] = \boxed{b\times [1 + r\times c]}$$

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The thing is: $r$ is the chance of landing a critical hit, and $[1-r]$ is the chance of landing a regular hit. So, for a large number of hits ($N$), $[1-r]\times N$ hits will be regular; and $r\times N$ hits will be critical. Therefore, the total damage will be:

$$T = [1-r]\times N \times b + r\times N \times b \times [1+c]$$

We get the average damage ($d$) by dividing it by the number of hits ($N$), which results in the given formula.

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Let's illustrate it with a numerical example. If $r=10\%$, $c=100\%$ and $b=1$, the critical hit deals $2$ damage, moreover, we will have $1$ critical hit and $9$ regular hits for each set of $N=10$ hits (in average). So, the total damage would be:

$$T = 9\times1 + 1\times 2 = 11$$

And the average damage would be:

$$d = \frac{T}{N} = \frac{11}{10} = 1 \times\left[1 + \frac{1}{10}\times 1\right] = b\times[1 + r\times c]$$

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The alternative formula that you've presented ($b\times r \times [1+c]$) is the average damage of the critical hits only (disregarding the damage dealt by regular hits). It would be the average damage if we assume that regular hits are misses that deal no damage at all.